行政院國家科學委員會專題研究計畫 成果報告
以階層式違約率模型探討叢聚違約現象及其在多資產信用 衍生性商品評價之應用
研究成果報告(精簡版)
計 畫 類 別 ： 個別型
計 畫 編 號 ： NSC 99-2410-H-011-008-
執 行 期 間 ： 99 年 08 月 01 日至 100 年 10 月 31 日 執 行 單 位 ： 國立臺灣科技大學財務金融研究所
計 畫 主 持 人 ： 繆維中
計畫參與人員： 博士班研究生-兼任助理人員：余欣庭 博士班研究生-兼任助理人員：李永新
報 告 附 件 ： 出席國際會議研究心得報告及發表論文
公 開 資 訊 ： 本計畫涉及專利或其他智慧財產權，2 年後可公開查詢
中 華 民 國 101 年 02 月 20 日
中 文 摘 要 ： 本研究延伸段錦泉教授所提出的階層式違約率模型，並討論 此型態的違約相關性如何影響信用衍生性商品價格及其所隱 含的違約相關性。在原始的模型中，共同事件發生時若不立 即造成違約，則此事件的效果便立即消失。有鑑於此，本文 提出的延伸模型假設共同事件一旦發生，則得以持續影響整 個資產組合一段較長的時間。在這段時間中，存活公司的違 約率會顯著地上升以反應共同事件的持續影響力。我們使用 馬可夫鏈來描述此一模型，並利用矩陣計算的方法求出多資 產違約的機率分配，並依此計算出信用違約交換和抵押債權 分卷的價格，及其中所隱含的相關性。經由一系列的數值分 析，結果顯示此持續效果的存在確實顯著地影響信用衍生性 商品價格和隱含相關性。對於今日信用市場實證上所見高風 險高相關性的現象，本研究所提出的模型也提供了一些適當 的詮釋角度。
中文關鍵詞： 馬可天鏈，信用風險，違約率，階層式違約率模型，信用違 約交換，抵押債權分卷，隱含相關性
英 文 摘 要 ： This paper provides an analysis of an extended version of Duan‘s hierarchical intensity
(HI) model for portfolio credit risk and discusses how this type of default correlation structure
infuences the prices of multi-name credit derivatives and the implied correlations. Following the original HI model where common credit events are used to generate correlation among defaultable entities, the proposed extended HI model allows the impacts from these common events to sustain for a longer while.
The default intensity of each entity will move up by a significant amount when the impact sustains, until some time later when the effect fades away. We show the proposed model can be formulated as a Markov chain for which the standard matrix computation techniques are applicable in their analyses. These computation methods are implemented in the numerical examples where the default probability distributions under the proposed model are converted the prices (spreads) of the concerned credit derivatives, and are in turn further converted to the implied compound and base correlations. From our results, it is
observed that the newly introduced parameters in regard to the sustaining effects play important roles
in the prices and implied correlations, and this provides more insights into today‘s credit markets with higher risk and higher correlation than earlier years.
英文關鍵詞： Markov chain, credit risk, default intensity, hierarchical intensity model, credit default swap, collateralized debt obligation, implied correlation
Markov Chain Modeling and Implied Correlation Analysis under the Extended Hierarchical Intensity Models for Portfolio Credit Risk
Daniel Wei-Chung Miaoa,∗, Steve Hsin-Ting Yub, Yung-Hsin Leec
a,b,cGraduate Institute of Finance, National Taiwan University of Science and Technology, Taipei, 106, Taiwan
Abstract
This paper provides an analysis of an extended version of Duan’s hierarchical intensity (HI) model for portfolio credit risk and discusses how this type of default correlation structure influences the prices of multi-name credit derivatives and the implied correlations. Following the original HI model where common credit events are used to generate correlation among defaultable entities, the proposed extended HI model allows the impacts from these common events to sustain for a longer while. The default intensity of each entity will move up by a significant amount when the impact sustains, until some time later when the effect fades away.
We show the proposed model can be formulated as a Markov chain for which the standard matrix computation techniques are applicable in their analyses. These computation methods are implemented in the numerical examples where the default probability distributions under the proposed model are converted the prices (spreads) of the concerned credit derivatives, and are in turn further converted to the implied compound and base correlations. From our results, it is observed that the newly introduced parameters in regard to the sustaining effects play important roles in the prices and implied correlations, and this provides more insights into today’s credit markets with higher risk and higher correlation than earlier years.
Keywords: Markov chain, credit risk, default intensity, hierarchical intensity model, credit default swap, collateralized debt obligation, implied correlation
1. Introduction
Modeling default correlation has been a central issue in the pricing problems of portfolio credit derivatives, and has become even more important since the consecutive financial crises happening in recent years. There are several major ways to model default correlation. Some are based on the structural default models that can be dated back to the classical work of Merton (1974) and Black and Cox (1976). In this class of models, default is defined to be the asset price for a company hitting a certain barrier. Zhou (2001) is a recent representative work
∗Corresponding author: Email addresses: [email protected]
that generalizes this type of models to multiple names. Another class of models known as the reduced-form models, or intensity models, assume the default arrival follows a Poisson process with stochastic intensity (a.k.a. default rate, hazard rate). Examples include Jarrow and Turnbull (1995), Lando (1998), Madan and Unal (1998), etc. Duffie (2005) provides a good review of the intensity models with a focus on the mathematically tractable models based on affine processes. Duffie and Gˆarleanu (2001) is an important study of extending the intensity models to multiple names. The default correlation models based on the above approaches are also known as the bottom-up models where the joint default behavior is an aggregation of the individual names. Recently there has been some attention on the so-called top-down models which specify the aggregate behavior directly without concerning the individual behavior. A representative example is Arnsdorf and Halperin (2008).
In the category of the intensity models, when they are extended to multiple names, there are many ways of introducing correlation. One approach is based contagion, such as Jarrow and Yu (2001), Davis and Lo (2001), Frey and Backhaus (2008), where an entity’s likelihood to default is affected by the occurrences of other entities’ defaults. In these models, it is usually harder to specify entities’ default intensities because of their dependence on other entities’ defaults, especially when the number of entities increases. The other approach to default correlation modeling is through the use of a common factor that influences all entities.
An important example is the model proposed by Duffie and Gˆarleanu (2001), where the individual default intensity is contributed by two components, individual and common. The latter portion itself is a stochastic process that is common to all entities and has the effect of lifting or lowering all intensities simultaneously. The ratio between the individual and common portions gives an idea how an entity’s default behavior depends on the common factor. More recently, Duan (2010) proposed a hierarchical intensity (HI) model which allows a more direct impact of the common factor on each individual entity’s default.
In this paper, we propose an extended version of Duan’s HI model and provide its analysis.
The extension is motivated by the fact that the common credit event in the HI model can only affect the entities at the time instant it arrives. If an arriving event doesn’t cause immediate default, then its impact disappears at the same time. In view of this lack, we propose to allow the common events to influence the portfolio longer. When their impacts sustain, all entities enter a riskier status with a heightened default intensity. This heightened intensity will remain until some time later the impact from the preceding event dies away, at which time it goes back to its normal level. By doing so, we make the model closer to reality and generate stronger default correlation.
We intend to use this model to investigate how the introduced effect will affect the prices of multi-name credit derivatives. Among these products, two popular types are basket credit default swaps (CDSs) and collateralized debt obligations (CDOs). The former usually contains 5∼15 names (there are 5 names in the example considered in Chiang et al (2007)) while the later is more standardized, typically containing 125 names. The basket CDS can be seen as a protection contract against the n-th default in a basket (portfolio). It gives payoff if the concerned n-th default event is triggered. The CDO considered here is also a protection contract which pays off when the concerned tranche starts to suffer loss (referring also to
Table 1: Market quotes for standard iTraxx CDO tranches
Jan 31, 2007 Jan 30, 2009 Jan 31, 2011
Tranche (αL, αH) 5 yr 7 yr 10 yr 5 yr 7 yr 10 yr 5 yr 7 yr 10 yr
1. Equity (0.00, 0.03) 1034 2576 4066 6428 6738 7041 2217 3903 4956
2. Mezz - E (0.03, 0.06) 41.59 111.75 328.86 1185.63 1151.88 1109.06 −377 319 1219 3. Mezz - M (0.06, 0.09) 11.95 33.09 85.89 606.69 650.56 694.75 −366 88 813 4. Mezz - S (0.09, 0.12) 5.6 15.43 39.73 315.63 350.5 410.63 69.13 183.5 270.4 5: Senior (0.12, 0.22) 2 5.27 14.06 97.13 104.5 112.75 30.06 86 121.5
Index (0.00, 1.00) 23 32 43 165 152 146 117 102 70
(αL, αH) are the attachment and detachment points of the CDO tranches. Spreads are expressed in basis points (bps). In the data for Jan 31, 2007 and Jan 30, 2009, only the first tranche (0.00, 0.03) is quoted with a fixed upfront spread of 500 bps. This quotation convention has changed in the data for Jan 31, 2011, where all the 5 tranches are upfront with fixed spreads of 500, 500, 300, 100, 100 bps, respectively.
This is the reason of seeing negative spreads in the Jan 31, 2011 data. (Source: Markit website. These data are downloadable from: http://creditfixings.com/CreditEventAuctions/itraxx.jsp.)
Hull and White (2006), Hull and White (2010)). Under our extended HI model, the focus of these pricing problems is how the parameters of the sustaining effect will influences the prices (spreads, expressed in basis points).
In addition to investigating the prices of these multi-name credit derivatives, it is worth studying the default correlation implied from their prices. As commented in Hull and White (2008), “empirical research shows that as the default environment worsens, default correlation increases.” It is then of interest to see what today’s level of default correlation is, and whether our proposed model helps in capturing the high correlation environment. The standard way to measure default correlation of a large portfolio is based on the one-factor Gaussian copula model proposed by Li (2000). In this model, suppose that the marginal default behaviors have been sufficiently characterized, the copula correlation ρ is the only unknown parameter. The implied correlation ˆρ is thus defined to be the copula correlation such that the copula model gives the same price as observed from the market, i.e. (also see Hull and White (2006))
scopula( ˆρ) = smarket
where scopula(ρ) is the spread given by the copula model as a function of copula correlation ρ.
The situation is similar to the Black-Scholes model for European option pricing, where the volatility is the only unknown parameter. The implied volatility is defined to be the value ˆσ such that cBS(ˆσ) = cmarket where cBS(σ) is the European option price under the BS model as a function of volatility σ.
To get a feeling of how default correlations have changed through the recent years when the global credit environment has deteriorated, below we provide the implied correlation data calculated from the iTraxx CDO tranches on Jan 31, 2007, Jan 30, 2009, and Jan 31, 2011. Note that the implied correlations for Jan 31, 2007 have been presented in Hull and White (2008) (Exhibit 3, p.15), here our results intend to be a sequel of their results.
This is meaningful because since their work was published, the world has been suffering
0 0.05 0.1 0.15 0.2 0.25 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
tranche detachment point
compound correlation
(a) Comparison of implied compound correlations
Jan 31, 2011 Jan 30, 2009
Jan 31, 2007
0 0.05 0.1 0.15 0.2 0.25
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
tranche detachment point
base correlation
(b) Comparison of implied base correlations
Jan 31, 2007 Jan 30, 2009
Jan 31, 2011
Figure 1: Comparisons of implied correlations from the market quotes for iTraxx CDO tranches on Jan 31, 2007, Jan 30, 2009 and Jan 31, 2011: (a) compound correlations; (b) base correlations.
a number of financial crises including the credit crunch of 2008 and the European sovereign debt crisis starting from 2009. The global default risk has undoubtedly increased significantly, and we wonder how the implied correlations have changed since then. Based on the market quotes for the iTraxx CDO tranches as shown in Table 1, we are able to compare the implied correlations across the three chosen time points. Figure 1 shows the results which include both the compound correlation and the base correlation. As explained in Hull and White (2008), the former is defined for a particular tranche and its curve usually exhibits a smile-like shape. By contrast, the latter is defined for all the tranches more junior than the concerned one and the curve usually displays a monotonically increasing trend. It can be observed the magnitude of the correlation levels, no matter compound or base correlations, have increased significantly since 2007. It is noticeable that the compound correlations of the middle tranches have grown up to a level around 0.7∼0.8 while the base correlations of the senior tranches have also increased to 0.6∼0.7.
In view of the changes in the implied correlations through the years, we would expect our extended model is better suited to the current environment and more capable of capturing the high spread and high correlation phenomena than the original model. The linking of model parameters to spreads and correlations require an analysis on the probability distribution of the number of defaults. To this end, we formulate the default sequence of our extended model as a continuous-time Markov chain, and use standard matrix computation techniques to calculate the default probability distributions. A series of numerical examples are provided to demonstrate the influences of the sustaining effect by showing how the spreads and correlations depend on the new parameters related to this effect. From our results we see that both the spreads and implied correlations increase significantly in the extended model compared with the original model. This means that the proposed model is not just intuitively appealing, it also provides further modeling flexibility and gives more insights into the influences of the sustaining effect.
This paper proceeds as follows. Section 2 introduces the HI model and its extension to the E-HI model. Section 3 shows how the proposed model is formulated as a Markov chain and how it is analyzed. Section 4 provides numerical examples to show how the default probability distributions, the derivatives prices and implied correlations are affected by the newly introduced model parameters. Finally Section 5 concludes this paper.
2. The Extended Hierarchical Intensity Model for Clustered Defaults
In this section, we first review the original version of Duan’s hierarchical intensity model and discuss its differences from other models. Then we present how it is extended to incorpo- rate the impacts of the sustaining effect.
2.1. A Brief Review of Duan’s Original HI Model
Consider a defaultable entity i in an economy. In Duan (2010), the original version of the hierarchical intensity model is defined as below:
dDi(t) = dNi(t) + χidNc(t) + ςidNg(t).
In the above model, Di(t) is the random process indicating whether the entity i has defaulted.
It is a counting process but the counting will be stopped when it goes from 0 to 1. (The process is stopped at t if Di(t−) = 0, Di(t) = 1, and thus dDi(t) = 1.) Ni(t) is also such a counting process (will stop at Ni(t) = 1) which represents the arrival of the default event triggered by entity i itself. Nc(t) is a normal counting process standing for the arrival of the common credit events that affect all the entities in the economy. Ng(t) is another counting process of the group-specific events which have impacts only on a particular group of entities (such as different market sectors). χi and ςi are Bernoulli random variables, which are used to describe whether an arriving common event (dNc(t) = 1) or group-specific event (dNg(t) = 1) may cause this entity i to default.
It is clear that the default correlation is introduced by the common and group-specific events. As Nc(t) and Ng(t) play similar roles, in this study we consider a simpler version without the group-specific term. This doesn’t cause much loss of generality in the original model, and helps to attribute the entity’s default to either internal reason (caused by the individual entity itself) or the external reason (caused by the credit event affecting the whole market). With this in mind, we consider a group of m defaultable entities, and assume the default behavior of each entity is governed by
dDi(t) = dNi(t) + χidNc(t), i = 1, · · · , m. (1) The above HI model can be parametrized by (λi, λc, pi), where λi, λc are the arrival rates of the Poisson counting processes, Ni(t) and Nc(t), and pi is the probability to see χi = 1 each time when dNc(t) = 1. In this specification, λc describes how often common events arrive and pi controls how likely the arriving common event causes entity i to default. We begin with this model based on which we build our extended HI model.
It is worth noting the difference between the above HI model and the famous Duffie and Gˆarleanu (2001) model because, in some sense, they take similar forms. In the Duffie-Gˆarleanu (DG) model, it is the individual default rate that is specified, i.e.
dλi(t) = dXi(t) + αidXc(t),
where λi(t) is the default rate (intensity) for entity i. It consists of two components, Xi(t), which is a jump-diffusion process specific to individual entity i, and Xc(t), which is another jump-diffusion process common to all entities. When the market becomes riskier, the common component Xc(t) may see a greater value or even see a jump. This will drive the default rates of all entities higher. The parameter αi determines how the change in the Xc(t) will be carried over to λi(t). The specification of Xc(t) as well as αi help generate the correlation among entities.
The HI model and the DG model have many properties in common. They both use individual and common portions to build up the default model. The roles played by Di(t), Ni(t), Nc(t), χi in the HI model is similar to those played by λi(t), Xi(t), Xc(t), αi in the DG model. But they are fundamentally different and should not be confused. In the HI model, the common portion Nc(t) is a counting process which contributes directly to the default of the concerned entity i. But in the DG model, the common portion Xc(t) contributes only to
the default intensity. It drives the intensity higher which in turn contributes indirectly to the occurrence of default.
2.2. The Proposed Extended HI Model
In the HI model as defined in (1), Nc(t) is the source of the correlated defaults between entities. Once a common event happens (dNc(t) = 1) but does not cause entity i to default at once (χi = 0, with probability 1 − pi), the impact of this event goes away immediately.
However, in reality when an influential common event happens, even if it doesn’t cause imme- diate defaults, it usually brings about serious impacts on individual entities. Those vulnerable entities which do not default at once tend to become even weaker and are likely to default later.
Motivated by the fact that the HI model in (1) is unable to capture this phenomenon, in this study we extend the HI model such that the sustaining impact of the common credit event may be taken into account. The way we model this sustaining effect is to define another random variable H(t). When the market is in a normal state, H(t) = 0. It will be driven to H(t) = 1 when the common event occurs. As long as the market is still under the influence of this common event, then H(t) = 1 continues to hold. Some time later when this sustaining effect dies away and entities no longer suffer from its lasting impacts, this variable goes back to its normal level H(t) = 0.
To model the heightened level of default risk when H(t) = 1, we introduce two more parameters δi and µ. Upon the arrival of a common event when H(t) goes from 0 to 1, we assume the individual default rate will increase from λi to λi + δi. Namely, δi is the enhancement level and is specific to each entity i = 1, · · · , m. We also assume the holding time of H(t) = 1 follows an exponential distribution with mean 1/µ. After this holding time coming to an end, H(t) changes back to 0 and the default rate returns to its original level λi. The reason of using the exponential assumption is for the convenience of formulating the default behavior as a Markov chain (see Section 3). Obviously, µ is a parameter that is common to all entities.
Figure 2 provides a graphical illustration of what are described above. It is not difficult to see the original HI model is a special case of the E-HI model when δi = 0 or µ = ∞ (1/µ = 0). The main purpose of this study is to investigate the effect that is brought by such an extension, specifically how the sustaining effects of common events will contribute to the prices of credit derivatives and default correlations. In the subsequent sections, we discuss the analysis of the proposed model.
3. Markov Chain Formulation and Analysis of the Proposed Models
For the pricing of portfolio credit derivatives, it is essential to calculate the probability distribution of the number of defaults, N (t), at any particular future time t, i.e.
P(N (t) = n), n = 0, 1, · · · , m, ∀t ≥ 0.
common credit event original default rate
back to normal
i i
heightened default rate
i i
exponential risky period with mean holding time 1
sustaining effect of the common event
( ) 1 H t
( ) 0 H t
Figure 2: The parameters δi and µ in the extended HI model.
Our approach is to formulate the original and extended HI models as continuous-time Markov chains, followed by using the standard techniques to solve the desired probabilities.
3.1. Modeling the Original HI Model
To illustrate, we start with the Markov chain modeling of Duan’s original HI model. In the homogeneous case, each default brings the same effect to the portfolio and hence only the number of defaults N (t) needs to be recorded. We may directly formulate the Markov chain for the state variable N (t) which is enough to describe the default sequence. Shown in Figure 3(a) is a simple example for m = 2. The role of the common event can be seen in all the state transitions. For example, for state changes from N (t) = 0 to N (t) = 1, it is either the case that one of the two entities defaults by itself (with rate 2λi), or the case that the arriving common event causes one to default but doesn’t affect the other (with rate 2λcpi(1 − pi)). If the arriving common event causes both entities to default at the same time (with rate λcp2i), then the state shall go from N (t) = 0 to N (t) = 2.
When the portfolio becomes heterogeneous, recording N (t) only is not enough as each entity behaves differently. To describe the individual behavior, let Di(t), i = 1, · · · , m, denote the indicator random variable of whether entity i has defaulted. Then the random vector (D1(t), · · · , Dm(t)) forms an m-dimensional Markov chain with N (t) = Pm
i=1Di(t). The Markov chain for m = 2 is given in Figure 3 (b). The necessity of differentiating the state (D1(t), D2(t)) = (1, 0) and (0,1) is clear: when one entity has defaulted, we need to know which the defaulted one is so that we may determine how the chain will switch to the next state (1,1) (because the transition rates are different). The transition rates involving λc and p1, p2 explain how the aggregate default behavior is affected by the common event. When the individual parameters are identical, the heterogeneous Markov chain will be reduced to the homogeneous Markov chain.
Figure 3: Markov chain formulation of the original HI model for the two-entity portfolio (m = 2): (a) homogeneous case with state variable defined as N (t); (b) heterogeneous case with state variable (vector) defined as (D1(t), D2(t)). Labeled inside each circle is the state variable.
3.2. Modeling the Extended HI Model: the Homogeneous Case
For the proposed extended model, the Markov chain is a bit more complicated. For convenience, we look at the homogeneous case first as the heterogeneous case is a further generalization. Recall that modeling N (t) is sufficient in the original HI model, here in the extended HI model, we need to incorporate H(t) and define the state variable of the Markov chain as (N (t), H(t)) = (n, h), n = 1, · · · , m, h = 0, 1. The purpose of the exponentially distributed holding time with mean 1/µ is such that we can describe the transition rate of H(t) = 1 → 0 by µ (in the same way as describing the transition rate of H(t) = 0 → 1 by λc). Depicted in Figure 4 is an example for m = 2. For a complete description of the Markov chain (N (t), H(t)), we need to specify all the transition rates among these states.
Let γ(n,h),(n0,h0) denote the transition rate form (n, h) to (n0, h0). Since N (t) must be a non- decreasing function of time, it is not possible to see a transition from (n, h) to (n0, h0) where n0 < n, i.e. γ(n,h),(n0,h0) = 0 for all n0 < n. We only need to consider those transitions with n0 ≥ n. In the following we divide these transitions into a few categories:
• Transitions between (n, 0) and (n, 1), n = 1, · · · , m (i.e. n0 = n):
γ(n,0),(n,1) = λc(1 − pi)m−n, γ(n,1),(n,0) = µ.
Figure 4: Markov chain formulation of the extended HI model with m = 2 homogeneous entities. The state variable is (N (t), H(t)).
The rate on the left says that such a transition happens when a common event arrives and it causes none of the remaining entities (numbered m − n) to default. The rate on the right is simply because it is the rate of H(t) coming back to normal if nothing else happens.
• Transitions from (n, 0) to (n + l, 0), l = 1, · · · , m − n:
γ(n,0),(n+l,0) =( (m − n)λi, l = 1;
0, l ≥ 2.
The transition rate is zero for l ≥ 2 because in normal state H(t) = 0, defaults happen independently according to their own default intensities. In this case, no simultaneous defaults may happen, hence the real transitions exist only when l = 1. The rate of seeing the next default is thus the sum of the intensities of all remaining (surviving) entities.
• from (n, 0) to (n + l, 1), l = 1, · · · , m − n:
γ(n,0),(n+l,1) =m − n
l
λcpli(1 − pi)m−n−l, l = 1, · · · , m − n.
All these rates are nonzero because such transitions are due to the common credit event.
When it actually comes, it may cause any number of the remaining (m − n) entities to default, and hence the formula of a binomial probability distribution is seen.
• Transitions from (n, 1) to (n + l, 0), l = 1, · · · , m − n:
γ(n,1),(n+l,0) = 0, l = 1, · · · , m − n.
This simply means that after the occurrence of a common event, the state variable H(t) should remain in H(t) = 1, and will not switch back to normal state 0 at the same time as the arrival of new default events.
• Transitions from (n, 1) to (n + l, 1), l = 1, · · · , m − n:
γ(n,1),(n+l,1) =
(m − n)(λi+ δi) + λcpli(1 − pi)m−n−l , l = 1;
m − n l
λcpli(1 − pi)m−n−l, l ≥ 2.
This is slightly more complicated. During the period that the sustaining effect is in presence (H(t) = 1), both the individually driven defaults and common event driven defaults may happen. Those driven by an individual entity can only cause a transition with l = 1, but those driven by a common event may cause a transition with all eligible l = 1, · · · , m − n.
As shown above, all the transition rates are now specified. For each state (n, h), we may further define Σ(n,h) to be the sum of its total outgoing rates, i.e.
Σ(n,h) = X
∀(n0,h0)6=(n,h)
γ(n,h),(n0,h0).
Then the generator matrix (usually called the Q matrix) can be clearly specified as given in Figure 5, meaning that the continuous-time Markov chain (N (t), H(t)) is well defined.
3.3. Modeling the Extended HI Model: the Heterogeneous Case
When it comes to the extended HI model for a heterogeneous group of entities, the sit- uation is more complicated. Combining the thoughts in the preceding two subsections, we shall formulate the Markov chain (D1(t), · · · , Dm(t), H(t)) which records whether each en- tity has defaulted and whether the previous credit event continues to influence. Let D(t) = (D1(t), D2(t), · · · , Dm(t)), the state variable is abbreviated to be (D(t), H(t)).
For convenience we further define kXk =Pm
i=1xi for a zero-one vector X = (x1, · · · , xm).
Let n = kdk stand for the number of defaults if D(t) = d = (d1, · · · , dm). The fact that a defaulted entity can not be changed back to survival status implies that γ(d,h),(d0,h0) = 0 if any element in the vector d0 − d is negative (note that each element d0i − di is either 0, 1 or −1). Namely, transitions may happen only when d0 − d is a zero-one vector and l = kd0− dk ≥ 0. Here l represents how many defaults happen in one transition, taking values of l = 0, 1, · · · , m − n. Below we list the formulas for the non-zero transition rates.
(0,0)(0,1)(1,0)(1,1)(2,0)(2,1)(3,0)...(m,0)(m,1)
−Σ(0,0)λc(1−pi)mmλimλcpi(1−pi)m−10
m 2
λcp2 i(1−pi)m−20...0λcpm i µ−Σ(0,1)0m
(λi+δi)+λcpi(1−pi)m−1 0
m 2
λcp2 i(1−pi)m−20...0λcpm i −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 00−Σ(1,0)λc(1−pi)m−1(m−1)λi(m−1)λcpi(1−pi)m−20...0λcpm−1 i 00µ−Σ(1,1)0(m−1)
(λi+δi)+λcpi(1−pi)m−2 0...0λcpm−1 i −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 0000−Σ(2,0)λc(1−pi)m−2(m−2)λi...0λcpm−2 i 0000µ−Σ(2,1)0...0λcpm−2 i −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
. . . . . . . . . . . . . . . . . . . . . . .. . . . . . .
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 0000000...λiλcpi 0000000...0(λi+δi)+λcpi −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 0000000...−λcλc 0000000...µ−µ
Figure5:Thegeneratormatrix(Qmatrix)fortheMarkovchain(N(t),H(t)).Thestatesarearrangedinalexicographicalorder, i.e.(n,h)=(0,0),(0,1),(1,0),(1,1),···(m,0),(m,1).ThenotationΣ(n,h)isthesumofallthetransitionratesfromthestate(n,h), n=0,1,2···,m,h=0,1.
• Transitions between (d, 0) and (d, 1) (i.e. d0 = d, there is no more default happening):
γ(d,0),(d,1) = λc
Y
∀i∈S
(1 − pi), γ(d,1),(d,0) = µ,
where S denotes the set of surviving entities. Note that there are m − n (= the size of set S) terms of (1 − pi) in the product.
• Transitions between (d, 0) and (d0, 0):
γ(d,0),(d0,0) =
( λi, ∀d : d0− d = 1i, l = kd0− dk = 1;
0, ∀d : l = kd0− dk ≥ 2.
In the above, 1i = (0, · · · , 1, · · · , 0) where 1 appears in the i-th position.
• Transitions between (d, 0) and (d0, 1):
γ(d,0),(d0,1)= λc
m
Y
i=1
qi, qi =
pi, di = 0, d0i = 1 (entity i becomes defaulted);
1 − pi, di = 0, d0i = 0 (entity i continues to survive);
1, di = 1, d0i = 1 (entity i has already defaulted).
Note that here l = 1, · · · , m − n. For a given l, there are m−nl such transitions.
• Transitions between (d, 1) and (d0, 0):
γ(d,1),(d0,0)= 0.
• Transitions between (d, 1) and (d0, 1):
γ(d,1),(d0,1)=
(λi+ δi) + λc m
Y
i=1
qi, ∀d : d0− d = 1i, l = kd0− dk = 1;
λc
m
Y
i=1
qi, ∀d : l = kd0− dk ≥ 2.
With the above transition rates specified, one may define the generator matrix Q for the Markov chain (D(t), H(t)). Since the total number of states is 2m+1 and the dimension of the Q matrix is 2m+1 × 2m+1, it is too large to show this matrix here. But it is not difficult to formulate such a matrix in a programming language such as Matlab. For illustrative purpose, we once again provide the state transition diagram for the Markov chain (D(t), H(t)) when m = 2. This is shown in Figure 5.
Figure 5: Markov chain formulation of the extended HI model with m = 2 heterogeneous entities. The state variable is (D1(t), D2(t), H(t)).
3.4. Probability distribution of N (t)
Remember that our objective is to obtain the distribution of N (t). To this end we need the calculate the state probabilities. Consider the homogeneous case, define
π(t; n, h) = P((N (t), H(t)) = (n, h)), then all these state probabilities form a vector as shown below
π(t) = [π(t; 0, 0) π(t; 0, 1) π(t; 1, 0) π(t; 1, 1) · · · π(t; m, 0) π(t; m, 1)].
Using the forward equations for the continuous-time Markov chain, we have π0(t) = π(t)Q,
which is solved by
π(t) = π(0)eQt. (2)
Assuming that all entities are survival in the beginning and the market is not under the influence of the former common event, the initial state is π(0) = [1 0 · · · 0]. Then the state probability vector at any future time t can be obtained by calculating the matrix exponential eQt. When π(t) is found, the desired probabilities are obtained by
P(N (t) = n) = π(t; n, 0) + π(t; n, 1), ∀n = 0, 1, · · · , m. (3)
This same procedure applies to the heterogeneous case, where we need to use the following formula
P(N (t) = n) = X
kdk=n
[π(t; d, 0) + π(t; d, 1)] , ∀n = 0, 1, · · · , m. (4)
3.5. Portfolio Credit Derivatives Pricing and Implied Correlations
When the probability distribution P(N (t) = n), ∀n is ready, we may proceed to calculate the prices of basket CDSs and CDO tranches, as well as the compound and based correlations implied from them. Below we follow Hull and White (2006), Hull and White (2008), and give a summary of the procedure to calculate these quantities.
The idea of pricing is to make the present values of the premium leg (the fees that the protection buyer has to pay) and protection leg (the payoff given to the protection buyer upon defaults) equal. Let t1, · · · , tk be the premium payment dates of the concerned derivatives, and let β(t) be the discounting factor, i.e. the present value of $ 1 received at time t. Assuming the interest rate is a constant r, we may write β(t) = e−rt. Consider an nth-to-default (ntD) basket CDS which provides payoff for the loss caused by the n-th default event (which happens with probability P(N (t) ≥ n)). Denote the spread as s per year. The present value of the premium leg is s(A + B) where (see Hull and White (2008) for details)
A =
k
X
j=1
h
(tj− tj−1)P(N (tj) < n)β(tj)i ,
and
B =
k
X
j=1
tj − tj−1 2
P(N (tj−1 ≥ n) − P(N (tj ≥ n)
β(tj−1+ tj
2 )
.
On the other hand, the present value of the protection leg (the expected discounted payoff of the protection contract) is
C =
k
X
j=1
P(N (tj−1≥ n) − P(N (tj ≥ n)
(1 − R) β(tj−1+ tj
2 )
,
where R is the recovery rate. The spread is then found from
s = C
A + B. (5)
The calculation of the spread for the CDO tranches is based on a similar idea but is slightly more complicated. Consider a principal of $ 1 subject to credit risk, and let EP(t) < 1 be the expected principal at time t. In a CDO tranche covering the loss for the attachment and detachment points (αL, αH), the expected principal is given by
EP(t) =
dmLe−1
X
n=0
P(N (t) = n) +
dmHe−1
X
n=dmLe
p(N (t) = n)
"
αH− n(1−R)m
αH − αL
#
, (6)
where mL = αLm
1 − R, mH = αHm
1 − R, and dxe is the smallest integer greater than x. When EP(t) is ready, the terms A, B and C are calculated using the following formulas:
A =
k
X
j=1
h
(tj − tj−1)EP(tj)β(tj)i ,
B =
k
X
j=1
tj− tj−1
2
EP(tj−1) − EP(tj)
β(tj−1+ tj
2 )
,
C =
k
X
j=1
EP(tj−1) − EP(tj)
β(tj−1+ tj
2 )
. Hence the spread s can be obtained accordingly.
The implied compound and base correlations are useful for measuring the default corre- lation in the real market or under a particular model. In this study we assume the portfolio default behavior follows our E-HI model and intend to calculate the correlation implied by this model. Consider the k-th CDO tranche (αkL, αkH) under the E-HI model with spread sE-HI(αkL, αkH). Further let scopula(ρ; αL, αH) denote the spread for the tranche (αL, αH) under the one-factor Gaussian copula model with copula correlation parameter ρ. The compound correlation is the particular value ˆρkc such that
scopula( ˆρkc; αkL, αkH) = sE-HI(αkL, αkH). (7)
The base correlation is concerned with the expected loss for the tranche (0, αkH) which may cover a number of real CDO tranches (from the first to the k-th ones). Let ELcopula(ρ; αL, αH) be the expected loss under the copula model, then the base correlation for the k-th tranche is the particular value ˆρkb such that
ELcopula( ˆρkb; 0, αHk) =
k
X
j=1
ELcopula( ˆρjc; αjL, αjH), (8)
where ˆρjc, αjL, αHj are the compound correlation, the attachment and detachment points for the j-th tranche, j ≤ k.
4. Numerical Examples
This section provides numerical examples to investigate how the correlation structure under the proposed E-HI model influences the default probability distributions, the derivatives prices, as well as the implied correlations. The focus is on the differences caused by the sustaining effect, and these will be seen via the two newly introduced parameters δi and µ.
4.1. Default Probability Distributions
In the two major types of multi-name credit derivatives, a basket CDS contract usually contains 5∼15 names while there are typically 125 names involved in a CDO contract. This makes the two cases of m = 10 and 125 particularly interesting. Also because the typical lives of these contracts range between 3∼10 years, we consider the two cases of t = 5 and t = 10 in our examples. Throughout this section we assume constant interest rate r = 3.5% and constant recovery rate R = 40%.
The reason for us to start with the probability distribution of the number of defaults, i.e.
P(N (t) = n), for all n = 0, 1, · · · , m, is because this is the main determinant in the subsequent pricing issues. Since the original HI model has introduced common credit events to generate default correlation, one may expect the two tails of the distribution (i.e. P(N (t) = n), as a function of n at a fixed t) should be heavier than those in the no correlation case. This is intuitive because we may think of the extreme case where all entities have perfect correlation in the sense that either they all survive or they default simultaneously. In this case, the probability mass concentrates on two points: P(N (t) = 0) (the left tail) and P(N (t) = m) (the right tail).
On top of the HI model, the study of P(N (t) = n) under our E-HI model focuses on how the distribution curves move when the impacts of common events sustain. More specifically, it aims to see how the heaviness of both tails and the peak are influenced by nonzero δi and 1/µ.
Figure 7 compares the distribution curves under the E-HI model and HI model for m = 10, 125 and t = 5, 10. When m = 10, for both models, we use the common parameter set (λi, λc, pi) = (0.01, 0.5, 0.1), while for the E-HI model, we add the extra parameters δi = 0.05, 1/µ = 2.0.
For contrasting purpose, we also provide the curves under the “No Correlation” (NC) model where each entity’s default is driven by an independent Poisson process with effective intensity λNC = λi+λcpi = 0.06. The parameter set for m = 125 is chosen in a similar way: the common parameter set is (λi, λc, pi) = (0.005, 0.25, 0.1) with additional parameters δi = 0.05, 1/µ = 4.0.
The NC model parameter is λNC= λi+ λcpi = 0.03.
It is observed in Figure 7(a)(b) that when m = 10, the NC model has the most centered distribution curves with the lightest tails as expected. By introducing correlation through common events, the HI model is able to generate slightly heavier tails (e.g. n ≤ 1 and n ≥ 6).
From the curves of the E-HI model, we observe that the presence of the sustaining effect not only makes the right tail much heavier, it also pulls the whole distribution curves toward the right, indicating that it is more probable to see many defaults. From (c)(d) where the distributions for m = 125 are shown, although the curves exhibit a relatively more irregular pattern, similar phenomena can be observed. It is thus clear that the sustaining effect of common events will make the situation even worse by providing a significantly heavier right tail.
In Figure 7 we assume homogeneity among entities, i.e. each entity has the same parame- ters and the correlation structure is symmetric. It is of interest to look at what may happen to the distribution if the entities are in fact heterogeneous with asymmetric correlation. Taking
0 2 4 6 8 10 0
0.05 0.1 0.15 0.2 0.25
n
P(N(t)= n)
(a) m=10, t=5
E−HI Model HI Model No Correlation
0 2 4 6 8 10
0 0.05 0.1 0.15 0.2 0.25
n
P(N(t)= n)
(b) m=10, t=10
E−HI Model HI Model No Correlation
0 20 40 60 80 100 120
0 0.02 0.04 0.06 0.08 0.1 0.12
n
P(N(t)= n)
(c) m=125, t=5
E−HI Model HI Model No Correlation
0 20 40 60 80 100 120
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
n
P(N(t)= n)
(d) m=125, t=10
E−HI Model HI Model No Correlation
Figure 7: Comparisons of the probability distributions of the number of defaults under the E-HI, HI and NC models: (a) m = 10, t = 5; (b) m = 10, t = 10; (c) m = 125, t = 5; (d) m = 125, t = 10.
the following homogeneous parameter set as a benchmark:
λi = 0.05, λc= 0.5, pi = 0.1, δ = 0.2, 1/µ = 1.0,
we then add heterogeneity on each of them (where possible). Note that only λi, pi, δi can be made heterogeneous because they are associated with an individual entity. On the contrary, λc, 1/µ are parameters common to all entities and can not be made heterogeneous. In the next example, we consider a heterogeneous group of m = 10 entities which are divided into two subgroups. Each subgroup consists of 5 homogeneous entities with identical parameters, but the average parameters across the two subgroups are kept the same as the above benchmark.
Table 2 shows our design of the heterogeneous parameters. For each of λi, pi, δi, there are two levels of heterogeneity: level 1 (2) assumes the concerned parameter in the first subgroup is 50% (90%) lower than the benchmark value while it is 50% (90%) higher in the second subgroup. We intend to see the changes caused by the heterogeneity added in each of the individual parameters.
